Related papers: Properly stratified algebras and tilting
In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…
We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…
We extend the classical notion of standardly stratified $k$-algebra (stated for finite dimensional $k$-algebras) to the more general class of rings, possibly without $1,$ with enough idempotents. We show that many of the fundamental…
In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived…
We discover a class of projective self-dual algebraic varieties. Namely, we consider actions of isotropy groups of complex symmetric spaces on the projectivized nilpotent varieties of isotropy modules. For them, we classify all orbit…
Among the finitely generated modules over a Noetherian ring R, the semidualizing modules have been singled out due to their particularly nice duality properties. When R is a normal domain, we exhibit a natural inclusion of the set of…
In this article, we realize skew-gentle algebras as skew-tiling algebras associated to admissible partial triangulations of punctured marked surfaces. Based on this, we establish a bijection between tagged permissible curves and certain…
A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…
We give a proof, based on the rigidity of tilting complexes, that the class of self-injective finite-dimensional algebras over an algebraically closed field is closed under derived equivalence.
Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra {\Lambda}, we give bijections between two-term tilting complexes in Kb(proj {\Lambda}),…
We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally…
We identify a class of symmetric algebras over a complete discrete valuation ring $\mathcal O$ of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group…
The most impressively prolific exploration of superstring models (aiming for our physical reality) has been focused on worldsheet-supersymmetric gauged linear sigma models and the closely associated complex-algebraic toric geometry. Mirror…
We give new properties of algebras with finite Gorenstein dimension coinciding with the dominant dimension $\geq 2$, which are called Auslander-Gorenstein algebras in the recent work of Iyama and Solberg, see \cite{IyaSol}. In particular,…
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain…
Let $A$ be a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the…
In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a…
We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence…
We introduce the notion of homological systems $\Theta$ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional…
In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian…